Lie triple systems arose initially in Cartan's study of Riemannian geometry.
Jacobson [1]} first introduced Lie triple systems and Jordan triple systems in connection with problems from Jordan theory
and quantum mechanics, viewing Lie triple systems as subspaces of Lie algebras that are closed relative to the ternary product.
Lie-Yamaguti algebras were introduced by Yamaguti in [2]} to give an algebraic interpretation of the characteristic properties of the torsion and curvature of homogeneous spaces with canonical connection studied in [3]}. He called them generalized Lie triple systems
at first, which were later called “Lie triple algebras". Recently, they were renamed as “Lie-Yamaguti algebras". The notion of Lie-Yamaguti algebra is a generalization of both Lie algebra and the Lie triple system. A Lie-Yamaguti algebra is a vector space together with a binary and ternary operation satisfying some compatibility conditions. In [4]}, the authors studied \((2,3)\) -cohomology groups and related deformation theory of Lie-Yamaguti algebras.
